Studies in Non-Euclidian GeometryUniversity of Wisconsin--Madison, 1925 - 100 pages |
Common terms and phrases
absol absolute polars Actual line actual points allel angle of parallelism axis becomes Bolyai CHAPTER chewsky circle representation circles orthogonal CIRCLES WITH CENTER clidian coincides common perpendicular conic conjugate coördinates cos-¹ cuts the absolute diameter dist xy drawn elliptic geometry elliptic plane equation equidistant curves etry Euclid Euclidian geometry family of circles Figure finite fixed circle fixed point fundamental circle geom given circle given line Hence horocycle hyperbolic geometry ideal lines ideal points imaginary infinite number isotropic line kaua lels length line lies line repeated locus Mliptic non-Euclidian geometry orthogonal circles P₁₂ parabolic geometry paral parallel postulate paratactic points at infinity pole postulate Projective Geometry proper circles pseudosphere Q₂ quadrant radius equal required circle right angles ruled surface Saccheri sphere surface tangent Theorem tion tractrix true uniquely determined x²+ya+ XX₁ xypq
Popular passages
Page 1 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Page 47 - The sum of the angles in a triangle is less than 180 degrees, the more so, the larger the sides of the triangle.
Page 43 - If two parallel lines are cut by a transversal, the corresponding angles are equal.
Page 2 - Euclid's twelfth axiom, which is more properly speaking a postulate, was his starting point for proving that through a given point one and only one line can be drawn parallel to a given line.
Page 39 - Ex. 11. Two lines of unequal length bisect each other at right angles. Show that any point in either line is equidistant from the extremities of the other. (§ 54.) PROP. VII. THEOREM 59. From a given point without a straight line, but one perpendicular can be drawn to the line. (It follows from § 25 that, from a given point without a straight line...
Page 45 - ... of geometry (I do not speak of those of arithmetic) are merely disguised definitions. Then what are we to think of that question : Is the Euclidean geometry true? It has no meaning. As well ask whether the metric system is true and the old measures false ; whether Cartesian coordinates are true and polar coordinates false. One geometry can not be more true than another ; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient...
Page 2 - The Hypothesis of the Right Angle. (2) The Hypothesis of the Obtuse Angle. (3) The Hypothesis of the Acute Angle.
Page 43 - Let a and b be the two v given lines, which neither intersect nor are parallel. From any two points A and P on the line a, draw AB and PB' perpendicular to the line 6. If AB = PB', the existence of a common perpendicular follows from § 28. Therefore we need only discuss the case when AB is...
Page 19 - The additive property, .!•£•» distance ab + distance be *= distance ac. (2) The distance from a point to Itself is zero. (3) The distance between two points is unaltered by а translation of the line...
Page 26 - A circle is a conic which has double contact with the absolute circle, whose axis IB the chord of contact and whose center is the pole of the axis with respect to the absolute or the circle Itself.